The Relationship Between Private Profitability and Net Social Benefit

A project will generally have some direct benefits and costs that result
from the goods and services it produces and the resources it uses up. These
goods and services and resources generally will involve market transactions.
A project may also have external benefits and costs. These are the good
things and the bad things that result from the project and are imposed upon
society rather than resulting from market transactions. Thus the net scocial
benefit of a project can be expressed as:

The Benefits and Costs of the Direct Effects of a Project

For simplicity assume the project produces only one output, milk, and uses
only one input, labor.

Let Pw/o denote the price of milk without the project and
Pw the price with the project. The output and consumption of
milk without the project is denoted as Qw/o and with the project
as Qw. The output of the project is denoted as Qp and
hence

Qw = Qw/o + Qp.

The gross benefit of an increase in consumption of milk from Qw/o
to Qw is the area under the demand curve for milk from
Qw/o to Qw.
For a straight line demand function this
is equal to the area of the trapezoid shown and thus:

Gross Social Benefit = (1/2)(Pw/o + Pw)Qp

The gross social cost of the increased resource use due to the project is the
area under the supply curve for the resource over the range of resource use
from what it would be without the project to what it would be with the project.

The use of labor without the project is denoted as Lw/o and with
the project as Lw. The labor input of the project is denoted as
Lp and
hence

Lw = Lw/o + Lp.

Let Ww/o denote the wage rate for labor without the project and
Ww the wage rate with the project.
For a straight line supply function for labor this
is equal to the area of the trapezoid shown and thus:

Gross Social Cost = (1/2)(Ww/o + Ww)Lp

The revenue that accrues to the project is equal to the price of milk
with the project, Pw, times the output of the project,
Qw. The cost of labor for the project is the wage rate with
the project, Ww, times the labor used by the project, Lw.
Thus the profit of the project is given by:

Private Profit = PwQp - WwLp

The net social benefit of the project can therefore be expressed in terms
of private profit as follows: